Commitments, Intentions, Truth and Nash Equilibria

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1 Commtments, Intentons, Truth and Nash Equlbra Karl H. Schlag Unversty of Venna Péter Vda, Unversty of Mannhem February 7, 2014 Abstract We explore the common wsdom that games wth multple Nash equlbra are easer to play f players can communcate. We present a smple model of communcaton n games under complete nformaton wth two varants, dentfed by when communcaton takes place. Sendng a message before play captures talk about ntentons, after play captures talk about past commtments. We focus on equlbra where messages are beleved whenever possble, thereby develop a theory of credble communcaton. The connecton to credblty n sender-recever games under ncomplete nformaton s dscussed. Applyng our results to Aumann s Stag Hunt game we fnd that communcaton s useless f talk s about commtments, whle the effcent outcome s selected f talk s about ntentons. Ths confrms ntuton and emprcal fndngs n the lterature. Keywords: Pre-play communcaton, cheap talk, credblty, coordnaton, sender-recever games. JEL Classfcaton Numbers: C72, D83. We would lke to thank Attla Ambrus, Stefano Demchels, Olver Gossner, Takakazu Honryo, Andras Korna, Tymofy Mylovanov, Martn Petz, Larry Samuelson, Joel Sobel and Thomas Troeger for useful comments. Péter Vda receved fnancal support from SFB/TR 15 whch s gratefully acknowledged. Unversty of Venna, Department of Economcs. E-mal: karl.schlag@unve.ac.at Correspondng author. Unversty of Mannhem, Department of Economcs, L7,3-5, D Mannhem, Germany. E-mal: vdapet@gmal.com 1

2 1 Introducton Game theory s agnostc about how to play n games that have multple Nash equlbra. In many economcally relevant settngs, for example n entry, nvestment and votng decsons, multplcty of Nash equlbra s rather the rule than the excepton. Belefs can be mutually self-confrmng when all beleve that others focus on an neffcent equlbrum even f there are alternatve Nash equlbra where all are strctly better off. Yet, t s commonly beleved that neffcent equlbra wll not be played when players are allowed to communcate before they play a game. The reasonng s that t suffces that one player proposes an equlbrum outcome n whch all players are better off to upset belefs assocated to neffcent play (see for example [29]). At the same tme Aumann [1] clams that communcaton can be useless even n the smplest games, and llustrates ths nformally n a verson of the Stag Hunt game. Farrell [16] 1 objects and argues for ths game that t depends on when communcaton takes place. If communcaton occurs after the person communcatng has made a choce then he agrees. However, f communcaton occurs before makng a choce then he argues that communcaton wll lead all players to hunt the stag. Charness [8] runs experments for ths game that renforce the ntuton of Farrell. Smply addng cheap talk wll not reduce the set of equlbrum outcomes. A necessary condton for upsettng belefs supportng an neffcent equlbrum s that alternatve proposals can be made. These would be ntated by sendng unantcpated messages, naturally accompaned by an explanaton of the crcumstances surroundng the new proposal. One would also explan whch messages one would have sent f one had other ntentons or the crcumstances would be dfferent. For communcaton to then be successful the partes nvolved, both those that talk and those that lsten, must be able and wllng to rethnk ther ntentons. At a more basc level, messages have to be understood and beleved f these should have an mpact. We present a formal model of communcaton n two person normal form games that s able to capture these aspects of communcaton and use standard game theoretc analyss to understand the effects of communcaton. The nnovaton of ths paper, n terms of modellng communcaton, s that we focus on equlbra n whch messages are beleved whenever possble, a concept we refer 1 Ths s based on earler personal communcaton on ths matter, see [16]. 2

3 to as credble communcaton. We choose an extremely smple communcaton protocol: player one sends a message and player two lstens. We consder two dfferent models for when ths sngle message s sent. In the frst varaton, player one sends the message before ether player has chosen an acton (referred to as Talk and Play, abbrevated by TP). In the second varaton, player one has already chosen an acton, unobserved by player two, when he sends the message to player two. After recevng ths message player two then makes hs choce (Play then Talk, PT). In both varatons the message s selected from a language. Ths language tself s chosen by one of the players, who we call the nterpreter, rght before the message s sent. A language s defned as a partton of player one s acton set, dentfyng a message wth a partcular subset of actons. Ths modellng choce emerges naturally when we specfy later how we model belefs. 2 There are two languages of partcular nterest. Complete communcaton refers to the language n whch each acton of player one s assocated to a message, whch means that the language s gven by the fnest partton of the set of actons. No communcaton ndcates the language that only has a sngle message, ths message s then equal to the entre set of actons of player one. Our soluton concept stpulates for each language that each message wthn the language s truthful and beleved by player two f ths s possble, otherwse all messages from ths language are gnored. Formally we take the followng approach. To be truthful n TP means that player one chooses an acton wthn the message he has sent. Truthfulness n PT means that player one sends the message whch contans the acton he has chosen. Player two beleves a message f her belef s supported wthn that message. All messages n a gven language can be beleved, n whch case the language s called credble, f there s a weakperfect Bayesan equlbrum of the communcaton game n whch that language s fxed such that player one always tells the truth and player two beleves each message of player one. So player one has to tell the truth no matter whch acton he has chosen (n PT) or whch message he has sent (n TP). All messages from a language are gnored f both players act as f ths language s equal to no 2 Modellng a language as a partton can be justfed as follows. Ever snce Saussure, language s conceptualzed as a dscrete set of arbtrary sgns wheren the meanng of each sgn s obtaned from ts opposton to other sgns. Thus, experence s parttoned nto dscrete categores whch receve arbtrary labels, a fact that s well captured by the standard system of modeltheoretc semantcs ([26]), wheren lngustcs sgns are translated nto formulas of logc that get nterpreted n dscrete models (compare also to [6]). 3

4 communcaton. The soluton concepts for our enlarged game wth communcaton n whch the language s chosen by the nterpreter are called TPE and PTE respectvely. Both TPE and PTE are weak-perfect Bayesan Nash equlbra of the enlarged games wth the followng propertes. If a credble language s chosen then player one tells the truth and player two beleves the message. If nstead the language chosen s not credble then both players act as f the language no communcaton was chosen. We now return to our motvatng queston, whether communcaton leads to effcency 3. Consder the Aumann s Stag Hunt game: Player two S R S 9, 9 0, 8 Player one R 8, 0 7, 7 The analyss of ths game reveals that communcaton helps players to coordnate on huntng the stag under TP but that t s useless, and hence unable to refne the set of Nash equlbrum outcomes, under PT. Consder TP. If player one says Stag and player two beleves t then player two plays Stag and hence player one plays Stag. If player one says Rabbt and player two beleves t then player two plays Rabbt and hence player one plays Rabbt. Hence complete communcaton s credble. No communcaton s always credble. Now no matter how players play when the language no communcaton was chosen, the nterpreter can choose the language complete communcaton send the message Stag and receve the payoff 9. In any TP equlbrum players go for the stag. On the contrary, complete communcaton s not credble under PT. If player one has chosen Rabbt, and player two always beleves hm, then t s optmal for hm to le and to send the message Stag. Only no communcaton s credble and all three Nash equlbra of the underlyng game can be supported n a PTE. Ths result confrms the ntuton of Farrell [16] and the fndngs of Charness [8] and does not depend on whch player s assgned as the nterpreter (whch s not true n general). In partcular, communcaton does not necessarly lead to effcent outcomes under PT but t does n ths game under TP. 3 Coordnatng on the effcent equlbrum s a central assumpton n the theory of colluson, but the way n whch players can coordnate s not very well understood. Players may also need to coordnate on the effcent punshment. For a nce dscusson of these topcs see [35]. 4

5 Interestngly we also fnd that effcency need not result under TP. We present a smple 3 by 3 game of common nterest n whch the acton assocated to the unque effcent outcome s contaned n the support of any Nash equlbrum. The message that mples that player one wll not choose ths acton s not belevable. Consequently only no communcaton s credble and neffcent play can be supported (see also dscusson Secton 7.1). Yet n ths game we fnd that all PTE outcomes are effcent. Combnng ths result wth our results on Aumann s Stag Hunt game thus show that nether talk about ntentons (TP) nor talk about commtments (PT) s unambguously superor for nducng effcent equlbrum outcomes. The structure of the paper s as follows. Secton 2 ntroduces some basc notatons, the noton of languages and messages. In secton 3 we descrbe the TP game. In secton 4 we descrbe the PT game. In secton 5 we defne credblty of a language under TP and PT and defne our soluton concepts TPE and PTE and prove ther exstence. Secton 6 contans propostons about selectng Nash equlbra, suffcent condton for effcency and an example demonstratng the power of the nterpreter. We also present a smple 3 by 3 game n whch TPE does not yeld to effcent outcome but PTE does so. Secton 7 contans the dscusson. We gve a weaker verson of credblty under TP and show that t yelds to effcent outcome n the prevous example, but generally does not do so. We connect our noton of credblty under PT to the credblty noton of Rabn [28] by weakenng credblty under PT. We show examples n whch Rabn s [28] noton s too weak and n whch t s too strong compared to our defnton and dscuss a stronger verson of credblty under PT. We also dscuss a varant of our model, n whch the language choce s not modelled explctly and dscuss the power of the nterpreter and the power of the sender. Fnally, we dscuss the related lterature. Secton 8 concludes. 2 Prelmnares 2.1 The Underlyng Game Let Γ be a two player (player one (he), player two (she)) smultaneous move game wth fnte acton sets S j and von Neumann-Morgenstern utlty functons defned by the Bernoull utltes u j : S 1 S 2 R for player j = 1, 2. For a fnte set X let X be the set of probablty dstrbutons over X and let 5

6 C (ξ) = {x X : ξ (x) > 0} be the support of ξ X. z R 2 s a Nash equlbrum outcome f there s a Nash equlbrum σ S 1 S 2 of Γ such that u j (σ) = z j for j = 1, 2. z s the favorte (pure) Nash equlbrum outcome for player j f there s no (pure) Nash equlbrum outcome z such that z j > z j. Formally, messages are elements of a partton L of S 1. Ths partton s called a language. Formally, a language s a subset L of the power set of S 1 whch does not contan the empty set and s 1 S 1,!m L such that s 1 m. The set of all languages s denoted by L. Languages wll be chosen by the nterpreter who s one of the two players. We allow for randomzng over languages, hence choces n L. A message from L s m L and L(s 1 ) L denotes the message whch contans the acton s 1. The degenerate language {S 1 } that contans a sngle element can be nterpreted as there beng no communcaton. At the opposte extreme, the language that contans only sngletons, so L (s 1 ) = {s 1 } for all s 1 S 1, may be nterpreted as complete communcaton. These two languages wll thus be referred to as no communcaton and complete communcaton. We consder two scenaros for when communcaton takes place. In frst talk then play player one frst sends a message to player two and then both smultaneously play Γ. In frst play then talk player one frst prvately chooses an acton n Γ and then sends a message to player two after whch player two chooses an acton n Γ. 3 Frst Talk Then Play We frst model communcaton that occurs before ether player chooses an acton. Frst the nterpreter chooses the language L. Then player one sends a message m from ths language L. Condtonal on the chosen language and the sent message player one chooses an acton whch s not observed by player two. Fnally player two chooses an acton. The above defnes the followng game, denoted by Γ T P for = 1, 2: 1. Player (the nterpreter) chooses a language L L and communcates t to the other player. 2. Player one sends a message m L to player two. 3. Player one chooses an acton s 1 (non-observable for player two) 4. Player two chooses an acton s 2. 6

7 5. Payoffs are realzed, where player j receves payoff u j (s 1, s 2 ), j = 1, Let us denote by Γ T P (L) the game n whch L s gven and starts wth stage 3.1 The Strateges n Γ T P We now ntroduce the notaton for the possbly mxed strateges used n Γ T P. Let L be the mxed language choce of the nterpreter n stage 1, so L L. Gven language L L chosen by the nterpreter n stage 1 let m L 1 L be the mxed message sent by player one n stage 2 and let m 1 = (m L 1 ) L L. Let σ L 1 (m) be the mxed acton of player one n stage 3 after message m L has been sent n stage 2, so σ1 L : L S 1. Concernng player two, let σ2 L (m) be the mxed acton of player two n stage 3 gven the language L chosen by the nterpreter n stage 1 and the message m receved n stage 2, so σ L 2 : L S 2. We wrte σ j = (σ L j ) L L for j = 1, 2. Hence, a strategy profle n the game Γ T P s a tuple (L, m 1, σ 1, σ 2 ). 4 Frst Play then Talk In ths scenaro we model communcaton that takes place after player one has chosen an acton. It s analogous to Γ T P except the choce of player one s moved from stage 3 to stage 1. Consder the followng game, denoted by Γ P T for = 1, 2: 1. Player one chooses a mxed acton σ 1 S 1 and prvately observes ts realzaton, an acton s 1 C(σ 1 ). 2. Player (the nterpreter) publcly chooses a language L L. 3. Player one sends a message m L to player two. 4. Player two chooses an acton s 2 S Payoffs are realzed, where player j receves payoff u j (s 1, s 2 ), j = 1, 2. Let us denote by Γ P T (L) the game above n whch the nterpreter has to choose L n stage 2, that s L s fxed. 7

8 4.1 The Strateges n Γ P T Let σ 1 S 1 be the mxed acton of player one n stage 1. For = 1 let L 1 (s 1 ) be the mxed language chosen n stage 2 after acton s 1 has been realzed n stage 1, L 1 : S 1 L. If player two s the nterpreter then L 2 L s ndependent from σ 1. In stage 3, player one chooses a mxed message m L 1 belongng to the language L chosen n stage 2 gven that acton s 1 s the realzaton of σ 1 n stage 1, so m L 1 : S 1 L and m 1 = (m L 1 ) L L. In stage 4, player two chooses a mxed acton σ L 2 (m) that depends on the language L chosen n stage 2 and on the message m receved n stage 3, so σ L 2 : L S 2 and σ 2 = (σ L 2 ) L L. Hence a strategy profle n the game Γ P T s descrbed by (σ 1, L, m 1, σ 2 ). 5 Soluton Concepts In ths secton we frequently refer to the noton of weak-perfect Bayesan equlbrum ([24]). To fx notaton let µ L 2 (m) S 1 ndcate player two s belef about player one s acton after message m L. Let µ L 2 = (µ L 2 (m)) m L and µ 2 = (µ L 2 ) L L. 5.1 Credblty we defne the noton of credble lan- Before defnng equlbra n Γ T P guages under TP and PT. and Γ P T Defnton 1 We say that a language L s credble under TP f there s a weak-perfect Bayesan equlbrum (m L 1, σ1 L, σ2 L, µ L 2 ) of Γ T P (L) n whch player one always tells the truth, and player two always correctly antcpates player one s acton,.e., 1. for all m L, C(σ1 L (m)) m and 2. for all m L, µ L 2 (m) m, 3. for all m L, µ L 2 (m) = σ1 L (m). Remark 1 L s credble under TP f and only f there s a subgame perfect equlbrum (m L 1, σ1 L, σ2 L ) of Γ T P (L) n whch player one always tells the truth. Note 8

9 that condton 2 s superfluous, however we keep t to clarfy the role of condton 3, namely that we requre n addton to tellng the truth and belevng that player two always, and not just on the equlbrum path, correctly antcpates player one s acton (pont 3). See the weaker defnton 5 wthout pont 3 n the dscusson n secton 7.1. Defnton 2 We say that a language L s credble under PT f there s a weak-perfect Bayesan equlbrum (σ 1, m L 1, σ L 2, µ L 2 ) of Γ P T (L) n whch player one tells the truth, and player two beleves t,.e., 1. for all s 1 S 1, m L 1 (s 1 ) = L(s 1 ) and 2. for all m L, µ L 2 (m) m. 5.2 TPE We now present our equlbrum concept for TP. We search for a weak-pbe of Γ T P n whch communcaton s truthful and beleved when the language s credble, and where messages are gnored otherwse. Defnton 3 (TPE) (L, m 1, σ 1, σ 2, µ 2 ) s called a talk then play equlbrum (TPE) of Γ T P 1. L s determnstc and credble, f t s a weak-perfect Bayesan equlbrum of Γ T P and: 2. f L s credble then for all m L: C(σ L 1 (m)) m and µ L 2 (m) = σ L 1 (m) (truth-tellng and correctly belevng), 3. f L s not credble then: σ L j (m) = σ {S 1} j for all m L and j = 1, 2 (gnorance). Remark 2 An outcome s a TPE f and only f t s a subgame perfect equlbrum outcome of Γ T P wth truth-tellng for credble languages and gnorance for noncredble languages. 5.3 PTE Our equlbrum concept for PT s analogous to the one for TP. Communcaton s truthful and beleved for credble languages, otherwse all messages are gnored. 9

10 Defnton 4 (PTE) (σ 1, L, m 1, σ 2, µ 2 ) s called a play then talk equlbrum (PTE) of Γ P T f t s a weak-perfect Bayesan equlbrum of Γ P T 1. L s determnstc, ndependent of σ 1 and (ts values are) credble, and: 2. f L s credble then: for all s 1 S 1, m L 1 (s 1 ) = L(s 1 ) and for all m L, µ L 2 (m) m (truth-tellng and belevng), 3. f L s not credble then: σ L j (m) = σ {S 1} j for all m L and j = 1, 2 (gnorance). 6 Propostons It s easy to support the nterpreter s favorte Nash equlbrum of the underlyng game wth equlbrum language no communcaton, n partcular we obtan: Proposton 1 (Exstence) For any Γ and = 1, 2 there exsts a TPE and a PTE of Γ. It s smlarly straghtforward to show: Proposton 2 (Nash equlbrum) For any Γ the PTE (TPE) outcomes of Γ, n whch σ 1 (m 1 ) s pure, are Nash equlbra of Γ. Remark 3 If σ 1 (m 1 ) can be mxed then any PTE (TPE) outcome s n the convex hull of Nash equlbra. We sketch the proof. Player one may choose a mxed acton n PT and dependng on the outcome of hs randomzaton may send dfferent messages. Player two on the equlbrum path correctly beleves player one s acton, hence, t must be that Nash equlbra are played after the dfferent messages. Ths can be the case only f player one s ndfferent between the two (or more) Nash equlbra. Smlar argument shows that under what crcumstances would player one choose random messages on the equlbrum path n TP. Some qualfcatons about the games Γ for whch we state our followng propostons are needed. Gven Γ let NE(Γ) be the set of Nash equlbrum payoffs of Γ. When we say that a payoff profle or equlbrum s effcent we mean that there are no payoff profle n NE(Γ) whch (weakly) Pareto domnates t. Assume that there s a unque favorte (pure) Nash equlbrum of player one. 10

11 Proposton 3 (Class of games where all TPE are effcent) If Γ s supermodular and player one s favorte equlbrum s n pure strateges then all TPE are effcent f = 1. If the game s supermodularand exhbts dmnshng return and non-degenerate (see [4]) then all TPE are effcent f = 1. Proof: The frst part s straghtforward along the lnes of [25], [32]. All one has to show s that there s a credble language under TP and a message such that the unque equlbrum supported wthn that message s player one s favorte Nash equlbrum. Ths s the case f there s another equlbrum of the game such that ts support does not contan player one s favorte Nash equlbrum acton, or the game has a unque pure equlbrum. For the second part, [4] and [22] show that any mxed strategy equlbrum can have at most two actons n ts support gven dmnshng returns. It follows, that player one s favorte equlbrum cannot have both extreme pure Nash equlbrum n ts support hence there s a credble language wth a message contanng only the favorte Nash equlbrum of player one. The proposton above apples for example to games wth postve spll-over, or to Cournot wth lnear demand and t s not necessarly true for PTE. 4 We say that a game s self-choosng f for all s 1, s 1 t s true that u 1 (s 1, b 2 (s 1 )) u 1 (s 1, b 2 (s 1)), where b 2 : S 1 S 2 s player two s best response functon. Ths s weaker than Balga and Morrs s [3] noton of self-sgnallng: for all (s 1, s 2 ) S t s true that u 1 (s 1, b 2 (s 1 )) u 1 (s 1, s 2 ). We say that a game s of common nterest f for all (s 1, s 2 ), (s 1, s 2) S t s true that u 1 (s 1, s 2 ) u 1 (s 1, s 2) f and only f u 2 (s 1, s 2 )) u 2 (s 1, s 2). Common nterest games are self-sgnallng and self-choosng. Proposton 4 (Class of games where all PTE are effcent) If = 1 and 1. ether player one s favorte equlbrum s n pure strateges and the game s self-choosng, 2. or the game s self-sgnallng then complete communcaton s credble and all PTE are effcent and player one receves hs favorte Nash equlbrum payoff. 4 For example, n a game wth postve spll over and ncreasng best response, player one always wants to convnce player two that he has chosen hs hghest acton. Hence, n PT there cannot be no non-trval credble communcaton. 11

12 Proof: In self-choosng games complete communcaton s credble. In selfsgnallng games the favorte equlbrum of player one s n pure strateges. 6.1 A 3 by 3 Common Interest Game (TP) The game shown below demonstrates how communcaton can be useless n TP even f the game has common nterests because only no-communcaton s credble under TP; but t yelds to effcency n PT as complete communcaton s credble under PT. Player one Player two L N R T 5,5 0,0-3,-3 M -1,-1 1,1 2,2 B 4,4-2,-2 3,3 (T, L) s a pure strategy Nash equlbrum that leads to the unque effcent outcome. 5 It s natural that player one wants to say I wll play T. However, each of the other two Nash equlbra of ths game have T n the support of the correspondng equlbrum strategy of player one. 6 Ths means that player one cannot truthfully (n TP) communcate that she wll not be playng T. Consequently, only {{T, M, B}} s a credble language. Regardless of who s the nterpreter, nontrval nformaton about ntentons cannot be transmtted under credble communcaton n ths game. 7 Dscusson 7.1 Effcency n Common Interest Games wth Weak Credblty under TP Now we gve a weaker verson of credblty under TP (defnton 1) whch does not requre that player two guesses correctly player one s acton after recevng 5 In fact, T s self-commttng and the game satsfes self-sgnallng ([15], [17]). 6 The Nash equlbra of the examples are computed usng a program wrtten by Rahul Savan. The program s based on the algorthm descrbed n [2], and can be found at The other two mxed Nash equlbra τ and ρ are gven by τ 1 (T ) = 2/7, τ 1 (M) = 5/7, τ 1 (B) = 0, τ 2 (L) = 1/7, τ 2 (N) = 6/7, τ 2 (R) = 0 ρ 1 (T ) = 4/15, ρ 1 (M) = 43/60, ρ 1 (B) = 1/60, ρ 2 (L) = 4/15, ρ 2 (N) = 31/60, ρ 2 (R) = 13/60 wth correspondng outcomes 5/7 and 41/60. 12

13 out of equlbrum messages. Defnton 5 We say that a language L s weakly-credble under TP f there s a weak-perfect Bayesan equlbrum (m L 1, σ1 L, σ2 L, µ L 2 ) of Γ T P (L) n whch player one always tells the truth, and player two always beleves t. That s: 1. for all m L, C(σ1 L (m)) m and 2. for all m L, µ L 2 (m) m. Requrng only weak perfect Bayesan s weaker than subgame perfecton and hence allows for more credble languages. In the common nterest game n secton 6.1 {{T }, {M, B}} s weakly-credble. If player one says {M, B} player two can beleve t by puttng not too much weght on B and play R. Player one then s tellng the truth because he plays B. Hence weak-tpe yelds effcency f we allow ncorrect out of equlbrum belefs when defnng credblty under TP and n pont 2 of defnton of TPE. In fact, ths s true n general: Remark 4 In common nterest games weak-tpe are effcent. It can namely be shown that the language whch contans two messages {T } and S 1 \ {T } (where T s the acton of player one yeldng the best outcome) s weakly credble or the game has a sngle pure strategy equlbrum. 7.2 Ineffcency wth Weak Credblty under TP If we change the payoff (-3,-3) to (4,-3) after (T, R) n the common nterest game above we stll have multple Nash equlbra each contanng T n ts support. The game s stll of self-choosng hence PT yelds to effcent outcome. However, {{T }, {M, B}} s not weakly-credble under TP anymore. It s natural that player one wants to say I wll play T. But he cannot do so n equlbrum, because after the message {M, B} player two ether plays L or R no matter what he beleves n {M, B}. But then n both cases player one plays T whch s out of {M, B}. Ths means that player one cannot truthfully (n TP) communcate that she wll not be playng T. Smlarly after message M player two must beleve M and play R but then player one plays T. After message B player two must play L and then player one plays T. Consequently, only {{T, M, B}} s a weaklycredble language. Regardless of who s the nterpreter, nontrval nformaton about ntentons cannot be transmtted under weakly-credble communcaton n ths game. 13

14 7.3 Rabn s Credblty and Credblty under PT We compare our noton of credblty n PT to that of Rabn [28]. Rabn [28] defnes the noton of a Credble Message Profle (CMP) for smple communcaton games (sender-recever games) wth pror p over the types T of the sender. He does so by startng from a large enough message set M such that for each X T there s an exclusve set of messages M(X) M such that M(X ) M(X j ) = holds for all X X j. To smplfy exposton dentfy subsets of T wth messages, thus sendng X has the meanng that my type s n X. So the language s descrbed by the power set of T as opposed to a partton as n our case. Focus s on a subset of messages called a message profle X = {X 1,..., X D } where X X j =. Through defntons 1 tll 6 Rabn [28] defnes when X s a CMP. Broadly speakng, a message profle s a CMP f for each message belongng to X receved by the recever, gven that the recever beleves that she faces the types n the message, each type n the message gets hs best payoff. In partcular, messages wthn X are beleved even f they are sent by types outsde D =1X. Now consder PT as a sender recever game n whch the sender, player one can choose hs own type. We dentfy the sender wth player one, T wth S 1 and the recever wth player two. Rabn [28] nvestgates whch types can tell the truth, allowng others to le. Our approach however bulds on an understandng of communcaton n whch all types can be beleved. One reason s that types are endogenous n ths paper. We consder credblty of a sngle sender whle Rabn has many dfferent senders, dentfed by ther types. Hence, we only concentrate on CMP-s whch are parttons (languages n our sense) of T. Our defnton of credble languages uses equlbra of the enlarged games Γ P T (L) whch reles on dscplned belefs on the equlbrum path. Consder an alternatve defnton that does not refer to an equlbrum n whch a language s called credble f player two can form belefs wthn the messages such that no matter whch acton was chosen by player one, t s optmal for player one to tell the truth, gven that player two plays optmally gven her belefs. Clearly, f a language s credble under PT then t s credble under PT. Moreover, complete communcaton s credble f and only f t s credble. Belefs after messages contanng a sngle acton are fxed. Hence, one could hope that communcaton leads Nash equlbrum play when ths nvolves player one choosng a sngle acton. But player one may want to choose a dfferent acton and stll tell the truth (by sendng a vague message) and devate from the cand- 14

15 date equlbrum. It s easy to construct examples whch show that credbllty s too weak n the sense that player one can manpulate player two and acheve hs best (non equlbrum) payoff n the game. Non-exstence of equlbrum wth credble languages: Player one Player two L N R RR T 1, 1 0, 0 0, 0 0, 0 M 0, 1 3, 1 5, 0 0, 2 B 1, 1 2, 2 0, 0 3, 1 (T, L) s the unque Nash equlbrum of the game. But (T, L) cannot be the outcome of any PTE. The language L = {{T }, {M, B}} s credble under PT f we choose µ L 2 ({M, B}) = (α, 1 α) {M, B} so that σ L 2 ({M, B})(R) = 1 s a best response to ths belef, that s player two plays R optmally after message {M, B}. But then player one wll choose M nstead of T, send the message {M, B} and receve a payoff of 5. No other languages, but no communcaton s credble. However, {{T }, {M, B}} s not credble under PT. It follows that one must further restrct the set of credble languages. Rabn [28] offers a stronger 7 noton of credblty for sender recever games n terms of credble message profles, descrbed above. Hs defnton s clearly not applcable drectly to our settng because player one (the sender) can choose hs type. However, we mmedately have the followng observaton. If L s a CMP then t s credble. Notce also, that {{T }, {M, B}} s a CMP for an open set of prors n the example above and so CMP appears weaker than credblty under PT. Further nterestng comparsons can be made when consderng complete communcaton. In partcular, f complete communcaton s a CMP then t s also credble under PT. For self-sgnallng games complete communcaton s a CMP. There are games where complete communcaton s credble under PT but t s not a CMP (see Example 2 n [28]). Ths and the observaton above suggests that f player one can choose hs type optmally before communcaton takes place then t allows for more precse communcaton compared to standard sender recever setups. However, CMP s nether weaker nor stronger than credblty under PT 7 Rabn [28] argues that n some stuaton t s rather weak. Indeed, {{T }, {M, B}} s a CMP for an open set of prors. 15

16 (see the example above). Ths s because, the choce of acton n PT gves more possblty to communcate but the requrement that belefs must be correct on the equlbrum path (whch s not an ssue n CMP) restrcts the possbltes of credble communcaton. It s easy to fnd a condton whch guarantees that whenever a language s a CMP then the language s credble under PT, though we have found t too restrctve. Our framework gves nterestng possblty to analyze stuatons n whch player one wants to pool some of hs actons when playng a mxed equlbrum. In partcular, n Γ P T 1 we requre that the language s chosen optmally after each acton of player one. Mxng can be nterestng out of equlbrum as well, once we further restrct player two s out of equlbrum belefs for credble languages, for example by requrng the exstence of a proper 8 equlbrum of Γ P T (L) n whch player one tells the truth. 7.4 Communcaton-proof Equlbra Let us defne u P T (L) player s worst weak Perfect Bayesan equlbrum payoff n Γ P T (L) for some credble L C so that ths equlbrum satsfes pont 1 and pont 2 n defnton 2. Let us defne u P T = max L C u P T (L). Remark 5 For example u P T 1 equals player one s favorte Nash equlbrum payoff f there s a language L such that n any weak Perfect Bayesan equlbrum of Γ P T (L) n whch L becomes credble (satsfes pont 1 and pont 2 n defnton 2), (σ 1, σ L 2 (m L 1 (s 1 ))) for some s 1 C(σ 1 ) s hs favorte Nash equlbrum of Γ. For example, ths s the case n the battle of the sexes. Notce also, that n the battle of the sexes player 2 can avod the mxed equlbrum payoff by choosng complete communcaton. Let us smlarly defne u T P. Namely, let us defne u T P (L) player s worst subgame perfect equlbrum payoff n Γ T P (L) for some credble L C so that ths equlbrum satsfes pont 1, pont 2 and pont 3 n the defnton 1. Let us defne u T P = max L C u T P (L). Remark 6 For example u T P 1 equals player one s favorte Nash equlbrum payoff f there s a language L such that n any subgame perfect equlbrum of Γ T P (L) n 8 No other equlbrum concept has bte on out of equlbrum belefs n Γ P T (L). Properness n Γ P T (L) s very smlar to subgame perfecton n Γ T P (L). 16

17 whch L becomes credble (satsfes pont 1, pont 2 (and pont 3) n defnton 1), (σ L 1 (m L 1 ), σ L 2 (m L 1 )) s hs favorte Nash equlbrum of Γ. For example, ths s the case n the battle of the sexes. Notce also, that n the battle of the sexes player 2 can avod the mxed equlbrum payoff by choosng complete communcaton. Proposton 5 (The Power of the Interpreter and the power of the Sender) For any Γ f player s the nterpreter, n any TPE (PTE) player must get at least u T P (u P T (u P T ). Proof: After the choce of any credble L player must get at least u T P (L) (L)). Hence, player as the nterpreter (n any TPE or PTE) must get at least u T P (u P T ). The power of the nterpreter = 2: To demonstrate the power of the nterpreter n TP (PT) we exhbt an example where = 2 and player two, by choosng complete communcaton forces player one to communcate all the detals of hs choce, splts the support of the favorte (mxed) equlbrum of player one and player two gets her best payoff. Notce that u T 2 P = u P 2 T = u 2 ({{T }, {B}}) = 2. Player two Player one L N R RR T 1, 1 1, 1 2, 3 1, 2 B 1, 1 1, 1 1, 2 2, 3 Both languages are credble n TP (PT), player one s favorte equlbrum payoff s 0, obtaned by mxng equally lkely between T and B. But player two wll choose complete communcaton and get a payoff of 2 n all TPE (PTE). 9 We have explctly modelled the language choce n both setups and allowed to vary the dentty of the nterpreter. An alternatve s to be slent about how the choce of language s realzed and consder no communcaton as the status quo. Then, gven a putatve Nash equlbrum payoff z of the underlyng game, one could say that ths Nash equlbrum s not communcaton proof under TP (PT) gven player s the nterpreter f there s a credble language L such that u T P (L), (u P T (L)) s strctly hgher than z. It s because player could upset the equlbrum correspondng to z by offerng to communcate usng the language L. 9 We rule out equlbra, where player one babbles or always sends the same message gven that complete communcaton s credble. Blume [5] also rules out such equlbra by showng that poolng equlbra are not persstent n the sense of [19]. 17

18 Ths approach s followed by [33] for sender recever games. One could consder further varatons, for example one n whch players must agree to communcate through a gven language to upset a mutually bad equlbrum. 7.5 Related Lterature Farrell ([15], [17]) poneered the communcaton lterature n whch messages have an ntrnsc meanng. Typcally communcaton s about prvate nformaton, the stereotypcal model s a sender-recever game ntroduced by Crawford and Sobel [12]. In the lterature on neologsms, unexpected messages are checked n terms of ther credblty (self-sgnallng), wth reasonng becomng more nvolved when more than one message passes ths test (e.g. see [23]). [3] conduct a formal game theoretc analyss, thus avodng plausblty checks. In contrast to [3], we ncorporate choce of language and allow for partal nformaton revelaton. Moreover, under frst play then talk, prvate nformaton s endogenous. There are only few papers where communcaton s about ntentons and messages have meanng, as we model n frst talk then play. [16] nvestgates communcaton about ntentons n the lght of ratonalzablty, albet addng addtonal plausblty requrements. [27] formally analyzes elmnaton of weakly domnated strateges for a rch class of messages, provdng ntrcate condtons for rulng out messages that are opposte to each other. She fnds that a unque outcome s selected n Battle of Sexes but not n Aumann s Stag Hunt game, the latter result beng dffcult to nterpret. [18] frst treat ntentons as f they are prvate nformaton, requrng self-sgnallng, and then add a condton (self-commttng) that ensures that players behave accordng to ther ntentons. Accordng to our formalzaton, self-sgnallng s not relevant for communcaton about ntentons. [14] show for the level k model that there s always more coordnaton on pure Nash equlbra when there s one way communcaton. [13] consder evoluton n complete nformaton games, both for symmetrc games wth two-sded communcaton and asymmetrc games wth one-sded communcaton. Truth can be ncorporated n dfferent ways, as seen n the papers hghlghted above. Neologsms buld on nformal plausblty arguments. [3] restrct attenton to equlbra n whch all nformaton s transmtted. Other approaches nclude [9] who assumes that senders tell the truth wth postve probablty and [21] and [20] (see also [31]) where there s a cost of tellng a le. In our paper we assume that the recever beleves that the sender tells truth, provded ths s possble 18

19 under the gven language. Otherwse both behave as f there s a sngle message when truth-tellng trvally holds. In contrast to [3] ths also puts dscplne on out of equlbrum behavor. 10 There s a closely related paper by [34], albet where messages have no meanng, n whch a game wth multple selves s proposed to account for the fndngs of [8]. Informally t s clamed that a standard game-theoretc model wll not suffce. The focus s on sequental equlbra n whch nformaton s transmtted. These do not exst f the acton s chosen before the message s sent, but exst f the message s sent frst. Note that ths does not mrror the fndngs of [8], even f one assumes that players select among those equlbra n whch nformaton s transmtted. Ths s because neffcent equlbra exst n whch nformaton s transmtted when the message s sent frst. 11 There s also expermental evdence that addng one-sded pre-play communcaton ncreases effcency (see [10], [11], [7]). 8 Concluson Interestngly, despte the large lterature on communcaton n games, we seem to be the frst to use an equlbrum analyss to nvestgate the mpact of truthful communcaton under pre-play communcaton (as modelled n our frst talk then play scenaro). Truthful does not mean that players are forced to tell the truth. It means that the sender s able to convnce the recever whenever he can be beleved. 12 We call ths credble communcaton. Our fndngs show that effcency s not guaranteed n common nterest games that have more than two strateges per player. The debate rased by Aumann also necesstates that we present a model n whch communcaton occurs durng play, called frst play then talk. Ths model has ts own value as t s the frst step to understandng communcaton whle playng extensve form games of mperfect nformaton or mperfect montorng (see [30]). Results n the two models are very dfferent and are useful to hghlght how communcaton nfluences outcomes. They are 10 Note that [3] do not to consder the complete nformaton settng (talk about ntentons) as they fnd t dffcult to formalze ther ntutons n that context (see page 467 n [3]). 11 Let players coordnate on the mxed Nash equlbrum when message m s sent. If any other message s sent assume that they coordnate on the neffcent pure strategy Nash equlbrum. 12 Notce that, n the Stag Hunt game, f we would not requre that out of equlbrum messages must be beleved, then playng rabbt s an equlbrum as well. Ths s because after message Stag, player two can beleve that t means Rabbt, hence sendng the message Rabbt and playng rabbt becomes an equlbrum. 19

20 both very tractable when analyzng specfc games and can help understand n applcatons whch equlbra have good propertes. After all, partes wll typcally communcate and ths should be consdered formally when makng predctons, nstead of usng t only as a motvaton lke n the lterature on renegotaton. Clearly communcaton as modelled n ths paper s very specfc. Once our modellng approach s well receved we beleve t to be mportant to tackle varous extensons. We fnd t valuable, thereby contrastng the modellng of [3], to allow for general messages and to dentfy all equlbra wth truth-tellng, and not just those where all nformaton s transmtted. In other words, we wsh to predct outcomes n games, not to understand when all nformaton can be transmtted. Other extensons that are easy to mplement nclude consderng the case where player two s uncertan about whether or not player one has already commtted to an acton and consderng an n player game where only player one communcates to the others. For example, n a votng setup, TP could be nterpreted as publc announcement of the result of a survey about ntentons and PT as publc announcement of ext poll data. Extensons that requre more thought n terms of makng the rght modellng choce nclude two-sded communcaton. References [1] R.J. Aumann, (1990), Nash-Equlbra are not Self-Enforcng, n Economc Decson Makng: Games, Econometrcs and Optmsaton (J. Gabszewcz, J.-F. Rchard, and L. Wolsey, Eds.), Amsterdam, Elsever [2] D. Avs, G. Rosenberg, R. Savan, and B. von Stengel (2009), Enumeraton of Nash equlbra for two-player games, Economc Theory 42,1, [3] S. Balga and S. Morrs (2002), Co-ordnaton, Spllovers, and Cheap Talk, Journal of Economc Theory 105, [4] U. Berger (2008), Learnng n games wth strategc complementartes revsted, Journal of Economc Theory 143, [5] A. Blume (1994), Equlbrum Refnements n Sender-Recever Games, Journal of Economc Theory 64,1,

21 [6] A. Blume (2002), Coordnaton and Learnng wth a Partal Language, Journal of Economc Theory 95,1, [7] A. Blume and A. Ortmann (2007), The effects of costless pre-play communcaton: Expermental evdence from games wth Pareto-ranked equlbra, Journal of Economc Theory 132, [8] G. Charness, (2000), Self-Servng Cheap Talk: A Test of Aumann s Conjecture, Economc Theory 33, [9] Y. Chen (2004), Perturbed Communcaton Games wth Honest Senders and Nave Recevers, Journal of Economc Theory 146, [10] R. Cooper, D.V. DeJong, R. Forsythe and T.W. Ross (1989), Communcaton n the Battle of the Sexes Game: Some Expermental Results, The RAND Journal of Economcs 20, [11] R. Cooper, D.V. DeJong, R. Forsythe and T.W. Ross (1992), Communcaton n Coordnaton Games, The Quarterly Journal of Economcs [12] V.P. Crawford and J. Sobel (1982), Strategc Informaton Transmsson, Econometrca 50, [13] S. Demchels and J.W. Webull (2008), Language, Meanng, and Games: A Model of Communcaton, Coordnaton, and Evoluton, Amercan Economc Revew 98, [14] T. Ellngsen and R. Ostlng (2010), When Does Communcaton Improve Coordnaton? Amercan Economc Revew 100, [15] J. Farrell (1986), Meanng and Credblty n Cheap Talk Games, Unversty of Calforna, Berkeley, Department of Economcs workng paper [16] J. Farrell (1988), Communcaton, Coordnaton, and Nash Equlbrum, Economc Letters 27, [17] J. Farrell (1993), Meanng and Credblty n Cheap Talk Games, Games and Economc Behavor 5,4,

22 [18] J. Farrell and M. Rabn (1996), Cheap Talk, The Journal of Economc Perspectves 10, [19] E. Kala and D. Samet (1984), Persstent Equlbra n Strategc Games, Internatonal Journal of Game Theory 13,3, [20] N. Kartk (2009), Strategc Communcaton wth Lyng Costs, Revew of Economc Studes 76, [21] N. Kartk, M. Ottavan, and F. Squntan (2007), Credulty, Les, and Costly Talk, Journal of Economc Theory 134, [22] V. Krshna (1992), Learnng n games wth strategc complementartes, HBS Workng Paper , Harvard Unversty. [23] S.A. Matthews, M. Okuno-Fujwara, and A. Postlewate (1991), Refnng Cheap-Talk Equlbra, Journal of Economc Theory 55, [24] A. Mas-Colell, M.D. Whnston, and J.R. Green (1995), Mcroeconomc Theory, Oxford Unversty Press. [25] P. Mlgrom and J. Roberts (1990), Ratonalzablty, Learnng, and Equlbrum n Games wth Strategc Complementartes, Econometrca 58,6, [26] R. Montague (1970), Unversal Grammar, Theora 36, [27] Pe-yu Lo (2007), Language and Coordnaton Games, unpublshed manuscrpt. [28] M. Rabn (1990), Communcaton between Ratonal Agents, Journal of Economc Theory 51, [29] M. Rabn (1994), A Model of Pre-game Communcaton, Journal of Economc Theory 63, [30] J. Renault and T. Tomala (2004), Communcaton equlbrum payoffs n repeated games wth mperfect montorng, Games and Economc Behavor 49,

23 [31] M. Serra-Garca, E. van Damme, and J. Potters (2013), Lyng about What You Know or About What You Do?, Journal of the European Economc Assocaton 11, 5, [32] C. Shannon (1990), An Ordnal Theory of Games wth Strategc Complementartes, Workng Paper Department of Economcs, Stanford Unversty. [33] I. Zapater (1997), Credble Proposals n Communcaton Games, Journal of Economc Theory 72,1, [34] R. Zultan (2012), Tmng of messages and the Aumann conjecture: a multple-selves approach, Internatonal Journal of Game Theory. [35] M.D. Whnston (2006), Lectures on Anttrust Economcs, The MIT Press. 23